Stackelberg leaderfollower game

1
Stackelberg -leader/follower game
2 firms choose quantities sequentially (1)
chooses its output then (2) chooses it output
then the market clears This changes the game
relative to Cournot competition - (2) can
respond to (1)s output, so that there is no need
for a conjecture - (1) can anticipate (2)s
reaction when it chooses its output Subgame
perfect equilibrium is typically used in this
setting - start at the end of the game and work
backwards - Since firm (1) moves first, take q1
as given, find (2)s best response q2 - then
back-up and consider (1)s choice q1 Firm
(2)
2
Stackelberg first subgame
Firm (1) solves a different problem, since it
knows how (2) will react to its choice.
3
Nash Equilibria
This is a non-credible threat but it satisfies NE
conditions - the strategies are complete
contingent plans - players are best
responding on the equilibrium path
4
First-mover advantage
Note that if we compute the SPNE, the leader is
always better off than in the simultaneous move
case (Cournot competition) Since (2) best
responds, (1) can get at least the simultaneous
move payoff by choosing the Cournot q. There is
a first-mover advantage in any game with
strategic substitutes. Defn Strategies are
strategic substitutes if best response functions
are downward sloping, and strategies are
strategic complements if best response functions
are upward sloping. In games of strategic
complements, first movers have a disadvantage
5
Policy game
Consider a policy game, where we show how
politicians can get voters to spend too much.
A politician proposes a policy change and then
a voter decides whether to accept or reject the
proposal. We should think of the voter as the
median voter the marginal person. Suppose
that the policy proposal involves setting some
number - possibly a level of expenditure on
education, for example - normalize so that the
voters preferred point is 0 - suppose the
current level of expenditure is below the voters
preferred point
Suppose that the politician has to play a pure
strategy, but that the voter can randomize.
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Payoffs
We are assuming that the politicians payoff is
increasing in the expenditure level - we do
not need to be any more precise than that. -
denote the expenditure level by x
The voter will - reject any policy proposal
that gives a lower utility (is further away from
0 than sq) - accept any policy that has a
higher utility - be indifferent between
accepting and rejecting any proposal with the
same utility as sq, and therefore might mix
- at sq and -sq any probability of
accepting is a best response
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The SPNE
Now back up to the politicians move. Since
the politician wants expenditure to be as high as
possible, the politician will propose the highest
level of expenditure that the voter is willing to
accept.
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Infinite NE